Roper Resonance and 1-+ Meson

Roper resonance: lattice calculations and its nature Is 1-+ meson a hybrid?

χ QCD Collaboration: A. Alexandru, Y. Chen, S.J. Dong, T. Draper, M. Gong, F .X. Lee, A. Li, K.F. Liu, X.F. Meng, N. Mathur, Y. Yang

INT, Aug. 2, 2012 Many Facets of Roper Resonance

Roper resonance (P11 in π N scattering, with I = ½, M = 1440 MeV, Γ ~ 300 MeV) was discovered in pion-nucleon partial wave analysis in 1963. πγ N and N has large contamination from Δ. Confirmation from (α, α’) scattering and J/ψ → NN π decay. (review by L. Alvarez-Ruso)

M. Ablikim, et al., PRL 97, 062001 (2006) Many Facets of Roper Resonance Theory: Quark potential model prediction is 100-200 MeV too high (Liu and Wong, 1983, Capstick and Isgur, 1986) Skyrmion can accommodate it as a radial excitation (J. Breit and C. Nappi, 1984 , Liu, Zhang, Black, 1984; U. Kaulfuss and U. Meissner, 1985) Suggestion as a pentaquark (Krewald 2000); as a member of the antidecuplet (Jaffe, Wilczek, 2003) Perhaps a hybrid (Barnes, Close, etc. 1983)  Lattice calculations

Quenched Lattice Calculations of Roper

page 4 Roper on the lattice

4 issues about lattice calculations: • Radial excitation or pentaquark state? • Dynamical fermions • Variation vs Bayesian fitting • Chiral dynamics Radial excitation? q 4q State? • Roper is seen on the lattice with three-quark interpolation field. • Weight : 2 2 |<0|ON|R >| > |<0|ON|N>| > 0 (point source, point sink)

∑ψ(x)

∑ON(x) ∑ψ(y) ∑ψ(z)

Point sink Wall source

<0|ON(0)|N> > 0

However, <0|ON(0)|R> < 0

1S 2S Bethe-Salpeter Wavefunction   r r +

* ORN = dr Ψ (r)ΨN (r) = 0 at non - relativist ic limit, ∫ R * ORN = dr Ψ (r)ΨN (r) ↑ as mq ↓ ∫ R Nucleon and Roper wavefunctions for mπ = 633 MeV

ORN = 0.30

NSTR, 2009, page 8 Roper and Nucleon Wavefunctions at mπ = 438 MeV

ORN = 0.59

NSTR, 2009, page 9 Dynamical Fermions Dynamical Fermions (Overlap on DWF Configurations)

• Improvement of nucleon correlator with low-mode substitution

Point source: mN = 1.13(14) GeV;

Z3 grid source: mN = 1.08(5) GeV; 2431×= 64 lattice with m 331 MeV, a = 1.73 GeV− π Z3 grid smeared source: mN = 1.14(2) GeV; 47 configurations Variation: mN = 1.16(1) GeV

NSTR, 2009, page 11 • Roper resonance from Coulomb wall source

2 3 4 22 mN =++ m01 cmπ cm 2 π(mixed)+c3 mππ ln( m /µ )+ ...

31− 24×= 64 lattice with mπ 331 MeV(sea), a = 1.73 GeV

NSTR, 2009, page 12 NSTR, 2009, page 13 Roper and Nucleon Wavefunctions at mπ = 438 MeV

ORN = 0.59

NSTR, 2009, page 14 Variation with 2 operators (10 – operator 1, no smearing, 23 – operator 2, 3 smearing)

Variation with wall and point sources? Glozman and Riska Challenge Phys. Rep. 268, 263 (1996)

Hyperfine Interaction of Quarks in Baryons • Color-spin • Flavor-spin C C F F • • • • λ1 λ 2 σ 1 σ 2 λ1 λ 2 σ 1 σ 2 • One-gluon exchange • Goldstone boson exchange

page 16 Evidence of η’N GHOST State in S11 (1535) Channel

η η -- - - W > 0

W<0

Lattice2005, page 17

Dynamical Fermions

η η

+

η and Π

NSTR, 2009, page 20 N* spectrum in LQCD & dynamical coupling

Lattice N* states (mπ=396MeV) Dynamics of P11-states: The bare state at ~1750 MeV through coupling to inelastic channels generates 2 poles below 1400 MeV. They are identified with the “Roper” resonance.

P11(1440)

pion mass D15(1675) Dynamical coupling D13(1700)

D13(1520)

S11(1650)

S11(1535)

LQCD finds states as predicted in SU(6)xO(3) N. Suzuki et al. (JLab/EBAC), R. Edwards, J. Dudek, D. Richards, Phys.Rev.Lett.104:042302,2010 S. Wallace, PRD84, 074508 (2011)

8/15/2012 V. Burkert, KITPC 2012, Beijing, China 21

Nature of Roper Resonance --- current understanding

• Roper is the radial excitation of nucleon with large couplings to Nη and Nπ. The real part of the Nη and Nπ loops pushes down the pole of radial excitation and the imaginary part gives the width of Roper, much the same way the Nπ coupling changes the Δ mass and gives rise of its width. • Issues with lattice calculations: – Variation vs Bayesian fitting: the size of the operators. – Chiral dynamics of the fermion action.

NSTR, 2009, page 22 Is 1-+ Meson a Hybrid? Y. Yang, Y. Chen, KFL, arXiv:1202.2205 Exotics:  Glueballs  Hybrids ( q q g )  Tetraquark mesoniums ( q q q q ) ; 4 pentaquark baryons () qq

How to identify glueballs and hybrids in experiments and lattice calculations and distinguish them from ordinary q q mesons? exotic quantum numbers, e.g. PC +− −+ −+ J = 0 ,1 ,2 Meson Interpolation Fields dim 3 ΨΓΨ dim 4 Ψ Γ D Ψ , GG dim 5 Ψ ΓΨ G , Ψ Γ DD Ψ , GDG dim 6 Ψ ΓΨ Ψ ΓΨ , GGG , GDDG , etc

Dim 3 operators Ψ ΓΨ do not generate exotic mesons with J PC = 1 − + , 0 − − , 0 + − , 2 + − but they can be produced with dim 5 ΨΓΨ G ops. they are hybrids. Experiments and Lattice Results

Expts on 1-+ :

 π1(1400), M= 1376±17 MeV, Γ=300±40 MeV  π1(1600), M=1653±17 MeV, Γ=225±38 MeV (?)

Lattice calculations (Quenched):  UKQCD (1997): 2.0(2) GeV  MILC (1997): 2.0(1) GeV, 2.1(1) GeV  Lacock and Schilling (1998): 1.9(2) GeV  MILC (2003): ~ 1.6 GeV  Adelaide (2004): ~ 2.4 GeV; ~1.6 GeV  HSC (2010)  J. Dudek (2011)    J = L + S, P = (−)L+1, C = (−)L+S are non-relativistic definitions! For example: Pauli Spinor Dirac Spinor   L =1, S =1, J = 0 ψσ ⋅∆ψ ΨΨ   L =1, S =1, J =1 ψσ × ∆ψ Ψγ iγ 5Ψ    L =1, S =1, J = 2 ψσ ⊗ ∆ψ Ψγ ⊗ DΨ  L =1, S = 0, J =1 ψ∆ψ Ψσ ijΨ

Lower component of Dirac spinor has a different parity from the upper one.  1      ip.x Ψfree ∝  σ ⋅p  χ e    E + m  1- + Meson Exotic:     J = L + S, P = (−)L+1, C = (−)L+S  P L = even  J S = 1  C = −  Cannot be q q meson   Yet dim 4 Ψ γ D Ψ is 1- + (B.A. Li, 1975)  4 i   Ψ → γ Ψ − P : (x) ~4 ( x)  C : Ψ → C Ψ  Exotic q q ?  Note: there is no dim 3 op. for 2++ 

It is produced with ΨΨγ ijD .

Content and Interpolation Field • One cannot always judge the content of a hadronic state by its interpolation field. For example

– the lowest state from ΨΨ is ηπ and ππ , not af00 and .

– the lowest P-wave state from χ N is πN not S11 . • The lowest pseudoscalars from GG is ηη and ' not glueb all. < 0|GG |ηη >< , 0| GG | >≥< 0|GG  | G >

– will need <0| ΨΨγη55 | ><ΨΨ , 0| γη | > >> < 0| ΨΨ γ 5 |G > to help decide which one is a glueball. (Cheng, Li and Liu, PRD 79, 014024 (2009))

Taida, 2006, page 28 Criteria for a meson to be a hybrid: Compare the matrix elements of both the dim 4 and dim 5 operators of 1-+ against other ordinary mesons, particularly the 2++  <Ψγ Ψ−+ > < −+ −− ++ +± ++ > • Dim 4 m.e. 0 |44DOi |1 << 0 | | 0 ,1 ,0 ,1 ,2

−+ −+ −− ++ +± ++ • Dim 5 m.e. <Ψ 0 |εγijk jBO k Ψ |1 > >> < 0 |5 | 0 ,1 ,0 ,1 ,2> Dim 3, 4 (D-type) and dim 5 (B-type) operators Non-relativistic Reduction - why no such operators in quark model

 Note: DD + is the q q center of mass momentum which is not a dynamical d.o.f. in the constituent quark model. Charmoniums 3 Anisotropic 12 x 96 lattice with Wilson action, β = 2.8, ζ =5, as= 0.138 fm. 1 ν = ~0.7048; ma mct a ζ 1 ε = ~ 0.352, 2ma

 DD+  −ε vcc = ~ vcc ~ 0.3 ; dx3ψγ D ψ   → dx3†ijk χ σ B φ; 2m ∫∫4 i NR.. m jk  DD− 3  3†εijk dxε ψΣ D ψ   → dxχσ B φ vcrev = ~ 0.3 ∫∫ijk j k NR.. m j k 2m

1 × ~ 0.027(3) vrel

1 ×=0.0058(4) 1 0.020(2) ma ×= vcc 0.018(1) 1 0++ ,1 ++ ,1 +− : Γ≈ m.e. D m.e. 2m • Leading NR reduction is reasonably good.

−+ 1 Mixing of operators with different 1 : D m.e. ≈ B m.e. • m dimensions seems to be small. 1 ε = ~ 0.352 2m Strangeness mesons

• D and B m.e. of 1-+ are comparable in size to those of other ordinary mesons. • No evidence for 1-+ in the charmonium and strangeonium regions to be hybrids. Exotic Quantum Numbers • NR reduction shows that 1-+ involves a center of mass motion of the qq pair.

• MIT bag model (Jaffe and Johnson, 1976; DeGrand and Jaffe, 1976)

+± 1 +− Ψ(2 )= (S1/2 P 3/2 PS 3/2 1/2) → cm motion for 2 2 ⇒ Spectrum is doubled. Simiarly (hamonic oscillator wf) ,

−± 1 Ψ(1 )= (1S1/2 2 S 1/2 2 SS 1/2 1 1/2 ) 2 • The `exotic’ q.n. can be accommodated by C =() − ls+ −+   J= Ll ++ s1 : Ll= = 1, s = 1 P= Ll ++1

Conclusion By examining m.e. of dim 4 and dim 5 operators of 1-+ against those of ordinary mesons, we find no evidence for it to be a hybrid in the cc and ss regions. NR reduction shows that it involves a center of mass AM of the qq pair. These `exotic’ q.n. are accessible in chiral quark models, bag models, flux-tube models, and QCD. To accommodate these q.n., the parity and AM rules need to be modified to

ls+   C=−( ) , J = L + l + sP , = L ++ l1